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There are two help screens for the Options tab for the three-way ANOVA dialog:

•A different page explains the multiple comparisons options.

•This page explains the graphing and output options.

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Graphing options

If you chose a multiple comparison method that computes confidence intervals (Tukey, Dunnett, etc.) Prism can plot these confidence intervals.

Effect sizes

Prism can calculate effect sizes for three-way ANOVA that quantify the magnitude of each main effect and all possible interactions. These measures help you understand not just whether effects are statistically significant, but how much of the variation in your data each factor and interaction explains. For three-way ANOVA, Prism reports effect sizes separately for all seven sources of variation: three main effects, three two-way interactions, and one three-way interaction.

% of total variation

This is the most intuitive effect size, expressing what percentage of the total variability in your data is explained by each source. In a three-way design, you can directly compare the relative importance of main effects versus interactions. For example, if one two-way interaction (AB vs CD) accounts for 33% of total variation while the row factor accounts for only 10%, this tells you that the interaction effect is more important than the main effect in explaining your data. The percentages across all sources of variation (plus residual error) should sum to 100%.

Eta-squared (η²) and partial eta-squared

Eta-squared is mathematically equivalent to "% of total variation" but expressed as a proportion (0 to 1) rather than a percentage. Partial eta-squared represents the proportion of variance explained by each effect after removing variance due to other effects in the model. For three-way ANOVA, partial eta-squared is particularly useful because it accounts for the increased model complexity. With seven sources of variation competing to explain your data, partial eta-squared provides a clearer picture of each effect's unique contribution. Partial eta-squared values are typically larger than eta-squared values and are more commonly reported in the literature for complex factorial designs.

These values can be calculated from the ANOVA table with the following formulas:

Cohen's f

Cohen's f is derived from eta-squared and provides standardized benchmarks for interpreting the magnitude of each effect. Cohen suggested these guidelines:

•Small effect: f ≈ 0.10 (1% of variance explained)

•Medium effect: f ≈ 0.25 (6% of variance explained)

•Large effect: f ≈ 0.40 (14% of variance explained)

Cohen's f is particularly useful for power analysis when planning future experiments with factorial designs.

Cohen's f can be calculated from eta squared or partial eta squared using the following formulas:

The value for Cohen's f that Prism reports can be obtained using the formula above with partial eta squared.

Interpreting effect sizes in three-way ANOVA

Three-way ANOVA effect sizes require careful interpretation due to the complexity of the design. Pay special attention to the pattern of effects across main effects and interactions. A significant three-way interaction suggests that the relationship between two factors depends on the level of the third factor, which can be more scientifically interesting than simple main effects. However, three-way interactions often have smaller effect sizes than two-way interactions or main effects, simply because they represent more specific, conditional relationships in the data.

When evaluating your results, consider both statistical significance and effect size magnitude. A small but statistically significant three-way interaction might be less important scientifically than a large two-way interaction, even if both are significant. The percentages of variation explained help you prioritize which effects deserve the most attention in interpretation and follow-up studies. Effect sizes are especially valuable for three-way designs because they help you navigate the complexity of seven different sources of variation and identify which relationships in your data are most important to understand.

Output

Choose how you want P values reported, and how many significant digits you need.